Sounds like you may not have power from the transformer. You can check this using a multimeter, by testing the voltage between the red R wire and the blue C wire. You should read somewhere around 24VAC, though depending on the system it could be anywhere from about 6VAC to 30VAC.
Most thermostats only use the batteries to power the thermostat itself, LED display, programming, date/time, etc. Thermostats with a C wire (like yours seems to have), sometimes only use the batteries to remember your settings during a power outage. The actual power for signalling heat/cool, is supplied by a transformer in the furnace itself (the red R wire). If there is no power on the R wire, the thermostat has no way to call for heat/cool. In this case you'll have to have an HVAC tech troubleshoot, and possibly replace the transformer.
You could use your multimeter to verify the furnace itself is getting power, though this is slightly more dangerous since you'll have to open the service panel on the furnace. This could put you in danger of electrocution, and should be avoided if you are not familiar with HVAC systems.
The only other thing I can think for you to check, is to look for a serviceman switch and make sure it's in the ON position. A serviceman switch is a switch on the feed line to the furnace, that allows servicemen to turn the power off to the furnace to service it. It should be located very close to the furnace, and within line of site of the furnaces service panel. If there is line of sight from the furnace service panel to the breaker panel that feed the furnace; or this is an old installation, you may not have a serviceman's switch.
Your best bet, is to simply call in an HVAC technician to service the furnace.
Acid cell and Alkaline Batteries function by a process of controlled corrosion of two different elements which creates a voltage potential and a current if the circuit is completed.
This process is in continuous motion from the day of manufacture, while it sits on the shelf in its packaging, while installed in the equipment during off/idle periods. It accelerates with higher discharge rate during use, and continues after the cell is unable to produce useful power to run the device
Alkaline batteries have a sealed container that serves as one of the electrodes. While it undergoes a lot slower corrosion than the case on the old Acid cells, it still corrodes. It is part of the process that allows the battery to function. You don't prevent it.
- Dead batteries should never be left in equipment, their process is nearly complete and the shell is close to breach.
- Live batteries have a date on them for a reason. Chemical reaction occurs even if the battery is live and able to run the equipment. Often with today's LED technology, for example, the batteries will corrode through and start leaking long before they become too weak to run the flashlight.
- Prevention is worth a pound of cure. Remove batteries from the device when not in use for extended periods of time. If it is part of a crash kit where you need batteries with the device at all times for emergency use, store them in a separate ziplock bag and check their dates periodically as well as for leakage.
- Don't mix different cells in a device which takes more then one cell. Alkaline cells tend to leak when a potential voltage is applied, i.e. when you try to charge them. This can happen in devices with more than one cell in series if some of the cells have less capacity than others. This is why one should never mix cells by type, brand or age in devices where the cells are connected in series (which is virtually all devices that take cells in the same compartment).
Best Answer
Eneloop batteries are NiMH which has a normal voltage of 1.2V vs. 1.5V for an alkaline batteries. Your thermostat uses two batteries so the Eneloops will generate 2.4V instead of 3V for fresh alkalines. At a worst case, the thermostat will not work (or report low batteries with fully charged batteries). Best case, the batteries will last for a short time.
As for specifying brand name alkalines, it's probably a partnership as @ratchetfreak suggested. They may also be trying to avoid customer dissatisfaction when cheap, off-brand batteries work poorly or last for a short time.